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Nilai limit as x rightwards arrow infinity of fraction numerator square root of 5 x plus 4 end root minus square root of 3 x plus 9 end root over denominator 4 x end fraction equals...

  1. 0

  2. 1 half

  3. 1

  4. 2

  5. 4

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator square root of 5 x plus 4 end root minus square root of 3 x plus 9 end root over denominator 4 x end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator square root of 5 x plus 4 end root minus square root of 3 x plus 9 end root over denominator 4 end fraction times 1 over x end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator square root of 5 x plus 4 end root minus square root of 3 x plus 9 end root over denominator 4 end fraction times limit as x rightwards arrow infinity of 1 over x end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator square root of 5 x plus 4 end root minus square root of 3 x plus 9 end root over denominator 4 end fraction times 0 end cell row blank equals 0 end table

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Terakhir diupdate 06 Juni 2021

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Pembahasan Soal:

Limittak hingga

Terdapat beberapa cara untuk menentukan limit menuju tak hingga dengan bentuk akar, seperti berikut ini:

limit as x rightwards arrow infinity of open parentheses square root of p x squared plus q x plus r end root minus square root of s x squared plus t x plus u end root close parentheses equals open curly brackets table attributes columnalign left end attributes row infinity row cell fraction numerator q minus t over denominator 2 square root of p end fraction end cell row cell negative infinity end cell end table close  

Diperoleh begin mathsize 14px style infinity end style apabila p greater than s, diperoleh fraction numerator q minus t over denominator 2 square root of p end fraction untuk p equals s, dan begin mathsize 14px style negative infinity end style untuk p less than s.

 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of open parentheses square root of 16 x squared plus 9 x minus 2 end root minus open parentheses 4 x plus 1 close parentheses close parentheses end cell equals cell limit as x rightwards arrow infinity of open parentheses square root of 16 x squared plus 9 x minus 2 end root minus square root of 16 x squared plus 8 x plus 1 end root close parentheses end cell row blank equals cell fraction numerator 9 minus 8 over denominator 2 square root of 16 end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 times 4 end fraction end cell row blank equals cell 1 over 8 end cell end table end style

Jadi, limit as x rightwards arrow infinity of open parentheses square root of 16 x squared plus 9 x minus 2 end root minus open parentheses 4 x plus 1 close parentheses close parenthesesequals 1 over 8.

Roboguru

Nilai dari

Pembahasan Soal:

Limit x → ∞ dari suatu fungsi bentuk akar dapat diselesaikan dengan menggunakan rumus:

xLimax2+bx+cpx2+qx+r=2abq,untuka=p

jadi, nilai limitnya yaitu


xlim(4x2+8x+34x(x+2))====xlim(4x2+8x+34x2+8x)2488400

Jadi, jawaban yang tepat adalah C.

Roboguru

Nilai dari  adalah ...

Pembahasan Soal:

Untuk menyelesaikan limit menuju tak hingga open parentheses x rightwards arrow infinity close parentheses, kita gunakan limit dasarnya yaitu,

limit as x rightwards arrow infinity of a over x to the power of n equals 0

dengan a bilangan real dan n bilangan asli, artinya kita harus mengarahkan bentuk limit tak hingga menjadi rumus dasar dengan cara :

  1. Buat fungsinya menjadi bentuk pecahan, jika bentuknya dalam akar mak akalikan dengan bentuk sekawannya (merasionalkan)
  2. Bagi variabelnya dengan pangkat tertinggi

Maka,

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 3 x cubed minus 9 x squared square root of x plus x square root of x minus 3 square root of x squared end root over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x cubed over denominator x cubed end fraction end style minus begin display style fraction numerator 9 x squared square root of x over denominator x cubed end fraction end style plus begin display style fraction numerator x square root of x over denominator x cubed end fraction end style minus begin display style fraction numerator 3 square root of x squared end root over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 square root of x over denominator x cubed end fraction end style plus begin display style fraction numerator 7 x squared over denominator x cubed end fraction end style minus begin display style fraction numerator 2 x cubed over denominator x cubed end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x cubed over denominator x cubed end fraction minus fraction numerator 9 x squared square root of x over denominator x cubed end fraction plus fraction numerator x square root of x over denominator x cubed end fraction minus fraction numerator 3 square root of x squared end root over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 square root of x over denominator x cubed end fraction plus fraction numerator 7 x squared over denominator x cubed end fraction minus fraction numerator 2 x cubed over denominator x cubed end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x squared times x to the power of begin display style 1 half end style end exponent over denominator x cubed end fraction plus fraction numerator x times x to the power of 1 half end exponent over denominator x cubed end fraction minus fraction numerator 3 x over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 times x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x to the power of 2 plus begin display style 1 half end style end exponent over denominator x cubed end fraction plus x to the power of 1 plus begin display style 1 half end style end exponent over x cubed minus 3 over x squared end style over denominator begin display style fraction numerator 5 x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x to the power of begin display style 5 over 2 end style end exponent over denominator x cubed end fraction plus x to the power of begin display style 3 over 2 end style end exponent over x cubed minus 3 over x squared end style over denominator begin display style fraction numerator 5 x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus 9 over x to the power of begin display style 1 half end style end exponent plus x over x to the power of begin display style 3 over 2 end style end exponent minus 3 over x squared end style over denominator begin display style 5 over x to the power of begin display style 5 over 2 end style end exponent plus 7 over x minus 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 9 over infinity to the power of begin display style 1 half end style end exponent plus x over infinity to the power of begin display style 3 over 2 end style end exponent minus 3 over infinity squared end style over denominator begin display style 5 over infinity to the power of begin display style 1 half end style end exponent plus 7 over infinity minus 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 0 plus 0 minus 0 end style over denominator begin display style 0 plus 0 minus 2 end style end fraction end cell row blank equals cell fraction numerator 3 over denominator negative 2 end fraction end cell row blank equals cell negative 3 over 2 end cell end table end style


Jadi, nilai dari limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction adalah negative 3 over 2.


Oleh karena itu, jawaban yang benar adalah D.

Roboguru

Tentukan nilai:

Pembahasan Soal:

Langkah-langkah limit tak hingga bentuk akar:

  1. Kali dengan akar sekawan.
  2. Bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut.
  3. Substitusi x mendekati infinity.

Langkah pertama (kali dengan akar sekawan):

Ingat: left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis equals a squared minus b squared.

begin mathsize 12px style limit as straight x rightwards arrow infinity of left parenthesis square root of straight x squared minus 3 straight x minus 4 end root minus straight x minus 1 right parenthesis equals limit as straight x rightwards arrow infinity of open parentheses square root of straight x squared minus 3 straight x minus 4 end root minus left parenthesis straight x plus 1 right parenthesis close parentheses times fraction numerator open parentheses square root of straight x squared minus 3 straight x minus 4 end root plus left parenthesis straight x plus 1 right parenthesis close parentheses over denominator open parentheses square root of straight x squared minus 3 straight x minus 4 end root plus left parenthesis straight x plus 1 right parenthesis close parentheses end fraction equals limit as straight x rightwards arrow infinity of fraction numerator open parentheses square root of straight x squared minus 3 straight x minus 4 end root close parentheses squared minus left parenthesis straight x plus 1 right parenthesis squared over denominator square root of straight x squared minus 3 straight x minus 4 end root plus straight x plus 1 end fraction equals limit as straight x rightwards arrow infinity of fraction numerator straight x squared minus 3 straight x minus 4 minus left parenthesis straight x squared plus 2 straight x plus 1 right parenthesis over denominator square root of straight x squared minus 3 straight x minus 4 right parenthesis end root plus straight x plus 1 end fraction equals limit as straight x rightwards arrow infinity of fraction numerator negative 5 straight x minus 5 over denominator square root of straight x squared minus 3 straight x minus 4 right parenthesis end root plus straight x plus 1 end fraction end style 

Langkah kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut):

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator negative 5 straight x minus 5 over denominator square root of straight x squared minus 3 straight x minus 4 end root plus straight x plus 1 end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator negative 5 straight x minus 5 over denominator square root of straight x squared minus 3 straight x minus 4 end root plus straight x plus 1 end fraction cross times fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style fraction numerator negative 5 straight x over denominator straight x end fraction end style minus begin display style 5 over x end style over denominator square root of begin display style straight x squared over straight x squared end style minus begin display style fraction numerator 3 straight x over denominator straight x squared end fraction end style minus begin display style 4 over x squared end style end root plus begin display style straight x over straight x end style plus begin display style 1 over x end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style negative 5 minus 5 over straight x end style over denominator begin display style square root of 1 minus 3 over straight x minus 4 over straight x squared end root plus 1 plus 1 over straight x end style end fraction end cell end table

Langkah kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut):

Ingat: limit as x rightwards arrow infinity of k over x to the power of n equals k over infinity to the power of n equals 0.

  table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator begin display style negative 5 minus 5 over straight x end style over denominator begin display style square root of 1 minus 3 over straight x minus 4 over straight x squared end root plus 1 plus 1 over straight x end style end fraction end cell equals cell fraction numerator begin display style negative 5 minus 5 over infinity end style over denominator begin display style square root of 1 minus 3 over infinity minus 4 over infinity squared end root plus 1 plus 1 over infinity end style end fraction end cell row blank equals cell fraction numerator begin display style negative 5 minus 0 end style over denominator begin display style square root of 1 minus 0 minus 0 end root plus 1 plus 0 end style end fraction end cell row blank equals cell negative 5 over 2 end cell end table

Jadi, Nilai begin mathsize 14px style limit as straight X rightwards arrow infinity of left parenthesis square root of straight x squared minus 3 straight x minus 4 end root minus straight x minus 1 right parenthesis equals negative 5 over 2 end style.

Roboguru

Tentukan nilai:

Pembahasan Soal:

Dengan menerapkan konsep limit fungsi mendekati takhingga.

Variabel begin mathsize 14px style straight x end style dengan pangkat terbesar dari begin mathsize 14px style fraction numerator square root of straight x squared minus 4 end root plus square root of straight x squared plus 3 end root over denominator square root of 9 straight x squared plus 5 end root minus square root of 4 straight x squared minus 6 end root end fraction end style adalah begin mathsize 14px style square root of straight x squared end root end style maka pembilang dan penyebut pada limit fungsi dibagi oleh begin mathsize 14px style square root of straight x squared end root end style. Sehingga diperoleh:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator square root of straight x squared minus 4 end root plus square root of straight x squared plus 3 end root over denominator square root of 9 straight x squared plus 5 end root minus square root of 4 straight x squared minus 6 end root end fraction right parenthesis end cell equals cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator begin display style fraction numerator square root of straight x squared minus 4 end root over denominator square root of straight x squared end root end fraction plus fraction numerator square root of straight x squared plus 3 end root over denominator square root of straight x squared end root end fraction end style over denominator begin display style fraction numerator square root of 9 straight x squared plus 5 end root over denominator square root of straight x squared end root end fraction minus fraction numerator square root of 4 straight x squared minus 6 end root over denominator square root of straight x squared end root end fraction end style end fraction right parenthesis end cell row blank equals cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator begin display style square root of fraction numerator straight x squared minus 4 over denominator straight x squared end fraction end root plus square root of fraction numerator begin display style straight x squared plus 3 end style over denominator begin display style straight x squared end style end fraction end root end style over denominator begin display style square root of fraction numerator 9 straight x squared plus 5 over denominator straight x squared end fraction end root minus square root of fraction numerator begin display style 4 straight x squared minus 6 end style over denominator begin display style straight x squared end style end fraction end root end style end fraction right parenthesis end cell row blank equals cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator begin display style square root of 1 minus 4 over straight x squared end root plus square root of 1 plus fraction numerator begin display style 3 end style over denominator begin display style straight x squared end style end fraction end root end style over denominator begin display style square root of 9 plus 5 over straight x squared end root minus square root of 4 minus fraction numerator begin display style 6 end style over denominator begin display style straight x squared end style end fraction end root end style end fraction right parenthesis end cell row blank equals cell left parenthesis fraction numerator begin display style square root of 1 minus 4 over infinity squared end root plus square root of 1 plus fraction numerator begin display style 3 end style over denominator begin display style infinity squared end style end fraction end root end style over denominator begin display style square root of 9 plus 5 over infinity squared end root minus square root of 4 minus fraction numerator begin display style 6 end style over denominator begin display style infinity squared end style end fraction end root end style end fraction right parenthesis end cell row blank equals cell fraction numerator begin display style square root of 1 minus 0 end root plus square root of 1 plus 0 end root end style over denominator begin display style square root of 9 plus 0 end root minus square root of 4 minus 0 end root end style end fraction right parenthesis end cell row blank equals cell fraction numerator begin display style 1 plus 1 end style over denominator begin display style 3 minus 2 end style end fraction end cell row blank equals 2 end table end style   

Jadi, Nilai dari Error converting from MathML to accessible text..

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